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6.003 Homework #14 This homework will not be collected or graded. It is intended to help you practice for the ﬁnal exam. Solutions will be posted. Problems 1. Neural signals The following ﬁgure illustrates the measurement of an action potential, which is an electrical pulse that travels along a neuron. Assume that this pulse travels in the positive z direction with constant speed ν = 10 m/s (which is a reasonable assumption for the large unmyelinated ﬁbers found in the squid, where such potentials were ﬁrst studied). Let Vm (z, t) represent the potential that is measured at position z and time t, where time is measured in milliseconds and distance is measured in millimeters. The right panel illustrates f (t) = Vm (30, t) which is the potential measured as a function of time t at position z = 30 mm. Axon 0 10 20 30 z [mm] 40 50 f(t) = Vm(30,t) f(t) = Vm(30,t) +50 0 -50 -100 0 2 4 6 Time [ms] 8 10 Part a. Sketch the dependence of Vm on t at position z = 40 mm (i.e., Vm (40, t)). Part b. Sketch the dependence of Vm on z at time t = 0 ms (i.e., Vm (z, 0)). Part c. Determine an expression for Vm (z, t) in terms of f (·) and ν. Explain the relations between this expression and your results from parts a and b. Vm (z, t) = 6.003 Homework #14 / Fall 2011 2 2. Characterizing block diagrams Consider the system deﬁned by the following block diagram: X + − + A Y − 3 2 1 2 a. Determine the system functional H = A A Y X. b. Determine the poles of the system. c. Determine the impulse response of the system. 6.003 Homework #14 / Fall 2011 3 3. Bode Plots Our goal is to design a stable CT LTI system H by cascading two causal CT LTI systems: H1 and H2 . The magnitudes of H (jω) and H1 (jω) are speciﬁed by the following straightline approximations. We are free to choose other aspects of the systems. |H(jω)| [dB] −20 −40 ω [log scale] 1 10 100 1000 1 10 100 1000 |H1 (jω)| [dB] 24 6 ω [log scale] a. Determine all system functions H1 (s) that are consistent with these design speciﬁ­ cations, and plot the straight-line approximation to the phase angle of each (as a function of ω). b. Determine all system functions H2 (s) that are consistent with these design speciﬁ­ cations, and plot the straight-line approximation to the phase angle of each (as a function of ω). 6.003 Homework #14 / Fall 2011 4 4. Controlling Systems Use a proportional controller (gain K) to control a plant whose input and output are related by R2 F = 1 + R − 2R2 as shown below. X + − K F Y a. Determine the range of K for which the unit-sample response of the closed-loop system converges to zero. b. Determine the range of K for which the closed-loop poles are real-valued numbers with magnitudes less than 1. 6.003 Homework #14 / Fall 2011 5 5. CT responses We are given that the impulse response of a CT LTI system is of the form A t T where A and T are unknown. When the system is subjected to the input x1 (t) 3 1 t −2 the output y1 (t) is zero at t = 5. When the input is x2 (t) = sin πt 3 u(t), the output y2 (t) is equal to 9 at t = 9. Determine A and T . Also determine y2 (t) for all t. 6.003 Homework #14 / Fall 2011 6 6. DT approximation of a CT system Let HC1 represent a causal CT system that is described by ẏC (t) + 3yC (t) = xC (t) where xC (t) represents the input signal and yC (t) represents the output signal. xC (t) HC1 yC (t) a. Determine the pole(s) of HC1 . Your task is to design a causal DT system HD1 to approximate the behavior of HC1 . xD [n] HD1 yD [n] Let xD [n] = xC (nT ) and yD [n] = yC (nT ) where T is a constant that represents the time between samples. Then approximate the derivative as dyC (t) yC (t + T ) − yC (t) = . dt T b. Determine an expression for the pole(s) of HD1 . c. Determine the range of values of T for which HD1 is stable. Now consider a second-order causal CT system HC2 , which is described by ÿC (t) + 100yC (t) = xC (t) . d. Determine the pole(s) of HC2 . Design a causal DT system HD2 to approximate the behavior of HC2 . Approximate derivatives as before: dyC (t) yC (t + T ) − yC (t) y˙C (t) = = and dt T y˙C (t + T ) − y˙C (t) d2 yC (t) = . 2 dt T e. Determine an expression for the pole(s) of HD2 . f. Determine the range of values of T for which HD2 stable. 6.003 Homework #14 / Fall 2011 7 7. Feedback Consider the system deﬁned by the following block diagram. X + − α + − a. Determine the system functional + Y po Delay Y X. b. Determine the number of closed-loop poles. c. Determine the range of gains (α) for which the closed-loop system is stable. 6.003 Homework #14 / Fall 2011 8 8. Finding a system a. Determine the diﬀerence equation and block diagram representations for a system whose output is 10, 1, 1, 1, 1, . . . when the input is 1, 1, 1, 1, 1, . . .. b. Determine the diﬀerence equation and block diagram representations for a system whose output is 1, 1, 1, 1, 1, . . . when the input is 10, 1, 1, 1, 1, . . .. c. Compare the diﬀerence equations in parts a and b. Compare the block diagrams in parts a and b. 6.003 Homework #14 / Fall 2011 9 9. Lots of poles All of the poles of a system fall on the unit circle, as shown in the following plot, where the ‘2’ and ‘3’ means that the adjacent pole, marked with parentheses, is a repeated pole of order 2 or 3 respectively. Im z × (×)2 (×)3 Re z × Which of the following choices represents the order of growth of this system’s unit-sample response for large n? Give the letter of your choice plus the information requested. a. y[n] is periodic. If you choose this option, determine the period. b. y[n] ∼ Ank (where A is a constant). If you choose this option, determine k. c. y[n] ∼ Az n (where A is a constant). If you choose this option, determine z. d. None of the above. If you choose this option, determine a closed-form asymptotic expression for y[n]. 6.003 Homework #14 / Fall 2011 10 10.Relation between time and frequency responses The impulse response of an LTI system is shown below. h(t) 1 5 t −1 If the input to the system is an eternal cosine, i.e., x(t) = cos(ωt), then the output will have the form y(t) = C cos(ωt + φ) a. Determine ωm , the frequency ω for which the constant C is greatest. What is the value of C when ω = ωm ? b. Determine ωp , the frequency ω for which the phase angle φ is − π4 . What is the value of C when ω = ωp ? MIT OpenCourseWare http://ocw.mit.edu 6.003 Signals and Systems Fall 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.